Rep. Gary Condit's lawyer Abbe Lowell told a press conference in Washington, D.C., late Monday that the congressman would allow police access to his apartment if they wished, provide them telephone and cellular phone records and would make his staff available to investigators to help in the search for missing intern Chandra Levy.

"The congressman will provide whatever additional information or material he can to police," Lowell said.

Lowell said earlier Monday that Condit, D-Calif., has already satisfied investigators' demands and that the police and the press should shift their attention to others possibly involved in Levy's dissapearance.

"Surely the time has come to focus less on Rep. Condit and more on the 99 other people police have identified who might be as helpful in providing information that could find Chandra," Lowell said.

But Levy's parents, Susan and Robert Levy, remain dissatisfied with Condit's confession that he had an affair with their daughter and want the Democrat to take a lie detector test, The Washington Post reported Monday.

"The family wants the comfort of knowing that the people who were closest to Chandra are giving complete and truthful information to investigators," said Michael K. Frisby, a spokesman for the Levys and their attorney, Billy Martin.

Lowell said that if the police ask about a lie detector test he would discuss it with Condit, although he doubts that polygraph tests are generally useful.

On Friday, Condit admitted to Washington, D.C., police that he had had a long-term romantic relationship with the missing federal intern, a police source told Fox News.

Condit met with investigators for a third time on Friday. Police requested the third meeting because they were "uncomfortable" with what they had learned in the first two interviews, according to Executive Assistant Police Chief Terrance Gainer.

"We challenged him for clarity. He provided the clarity and answered each one of our questions," Gainer said.

In his first two meetings with authorities, Condit only said he broke off his "close friendship" with 24-year-old Levy two days before she disappeared, police sources said.

Condit's lawyer Abbe Lowell explained Condit's earlier silence as an attempt to protect his family.

"He's a public figure who still believes that a public figure can have a private life," Lowell told CNN.

Lowell also said he had no reason to believe that Condit, who was elected to Congress in 1989, would not complete his term or run again.

"I have no reason to think that's on his brain, radar screen, purview, constellation," Lowell said on ABC's This Week. On CNN, Lowell added that he thinks Condit's constituents "would like him to serve, and he wants to continue."

The relationship between Levy, a former intern at the Bureau of Prisons, and her congressman has been under scrutiny since the Modesto, Calif., woman disappeared from her Washington apartment. Levy was last seen on April 30.

Friday's meeting occurred in Washington at an undisclosed non-police location, according to Gainer. Gainer also reiterated that Condit is not now, and has never been, a suspect in the disappearance.

Gainer also said that an article in the San Francisco Chronicle saying that a federal grand jury was expected to call on Condit to testify was "incorrect."

The Chronicle reported in its Saturday edition that a grand jury would permit Washington police and the FBI to subpoena records that may be relevant to the case, including Condit's cellular phone bills.

Gainer dismissed the report, saying, "there is no need to go to a grand jury if the people with whom we're dealing are cooperative, and the people with whom we're dealing [Condit and his wife] are cooperative."

The grand jury speculation came just two days after Linda Zamsky, Chandra's aunt, said her niece confided details about her secret affair with the 53-year-old congressman.

"Hi, Linda. This is Chandra. My internship is over," the missing intern said in a message left on Zamsky's answering machine the day before she vanished in Washington. "I'm planning on packing my bags in the next week or 10 days. Heading home for a while. Don't know what I'm going to do this summer. And I really have some big news or something important to tell. Call me. ..."

Zamsky said she was coming out with the information she gleaned in the several months that she was Levy's confidante because she was upset that Condit's staff members have repeatedly denied there was a romantic relationship.

"The Levy family is frustrated and outraged that Congressman Gary Condit and his associates have mischaracterized Chandra Levy's relationship with the congressman," Zamsky said in a public statement. "From my many conversations with her, it was clear, without a doubt, that they were involved in an intimate relationship.

On Thursday, Zamsky added more evidence about what she called the "bedroom encounters" of Levy and Condit when she released a wealth of details about her niece's feelings for the married congressman, including Levy's hopes that she would eventually marry Condit and bear his babies.

Levy first told Zamsky about her affair with Condit last fall, Zamsky said. She said the congressman gave Levy gifts of jewelry and chocolate, bought her plane tickets and spent his weekends with her, all under a veil of the strictest secrecy. Zamsky said Condit went to great lengths to maintain the cover that he and Levy were merely platonic friends, including elaborate ruses to keep neighbors from learning that Levy was visiting him at his Adams Morgan apartment, or to quell suspicion when the two took taxicabs together.

"So these are all little details that she had worked out with him, that kind of, you know, really show how serious this relationship was and how serious it was that it had to remain a secret," Zamsky said.

At first Levy refused to name her lover, saying only that he was an older man in government, "looking a little bit like Harrison Ford. And she said he was lean, good shape, worked out, very conscientious about his body for 53 years old," Zamsky said.

But Levy eventually slipped and mentioned Condit by name, Zamsky said.

"I asked her, 'How do you get in touch with him if it's so secretive, this relationship?' And she said ... she would also call his [office]. She said, 'And ... they'd answer, Gary Condit.' And that's how his name came out ... Congressman Condit's office. Congressman Gary Condit's office. That's how the name came out.

"And she goes, 'Oops.' She says, 'Oh, you didn't hear that, did you?' And I said, 'No.'"

Zamsky said Levy told her she remained monogamous to Condit and hoped she would marry Condit after five years and have his baby. Zamsky characterized those plans as wishful thinking on Levy's part, with no evidence that Condit had led the younger woman to believe any of it would happen.

In April, Levy talked about her relationship with her "boyfriend, my guy," in more serious terms, Zamsky said. The next time Zamsky would hear from her niece would also be the last — the recorded message on her machine in which Levy said she was returning to Modesto and promised "something important to tell."

Zamsky described Levy's account of the relationship in a 90-minute videotaped interview with police.

Zamsky's description of a heavy romance between the congressman and her missing niece came as law enforcement officers questioned Condit's wife, Carolyn.

Law enforcement officials have given no indication why they wanted to question Mrs. Condit, other than their repeated statements that they wanted to talk to anyone who might shed any light on the case.

Fox News' Rita Cosby contributed to this report.